Edges First

Edges first, or just EF, is a method where all edges of the cube are solved before the corners.
EF is not well suited for advanced speedsolving, the first step is very easy but the corners needs a unplesant number of moves to be solved after that in most cases. In FMC, when the time to look ahead is of no or little importance it is useful if many corners can be solved together with the edges, then it may be only one or possibly two corners cycles left in the end, that may be solved using insert algorithms.
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History
Although never as popular as Corners First and Layer by layer, Edges First solutions have been around for a long time and were explored early on in the history of the Cube. In his Notes on Rubik's Magic Cube (1979) David Singmaster mentions (p. 26) that mathematician Roger Penrose had created an Edgesfirst solution which "takes about 100 moves, but he hasn't analysed it in detail". Other mathematicians also worked on similar methods (see pages 29 and 40).
Singlealgorithm beginner method
 Solve the cross
 Solve 3 E layer edges
 Orient and (then) permute the last layer edges while solving the last E layer edge. This can be done intuitively
 Rotate the cube so you have at least 2 D layer corners in the D layer and on FUR another D layer corner
 Use RUR'U' (Sexy Move) and D setup moves to solve 3 corners of the D layer. Be sure to use the unsolved corner to fix the edges that may have been affected by the sexy move (if it was not done 3 OR 6 times)
 Turn the cube over so that the 3/4 solved layer is now on top and the unsolved corner is on FUR
 Again use RUR'U' and D setup moves to solve the rest of the corners
 Parity: If the D layer is solved but not the cube you have parity. Repeat sexy move untill the cube is solved exept for 2 corners
 Hold the 2 Corners in D layer and solve them using RUR'U'
The method only needs the sexy move as an algorithm, but it needs a lot of intuition and knowledge about the cube. As a speedsolving method though, its move count is way too high.
Philip Marshall method
The Philip Marshall Method is an Edges First 2algorithm method created by Philip Marshall in 1998. The solution steps are:
 Solve a cross.
 Solve 3 of the 4 middlelayer edges using U'RUR'.
 With the same U'RUR' algorithm solve the top face edges (automatically solving the last middlelayer edge).
 Use URU'L'UR'U'L moves to solve the top 4 corners.
 Use the same algorithm to solve 1 of the last 4 corners, then use conjugation (if necessary) and the same algorithm again to solve the last 3 corners in one go.
More efficient than the single algorithm method (Marshall claimed an average of 65 moves). Still a slow method though since a lot of time is spent of inspection.
1990s Rubiks.com solution
In the late 1990s/early 2000s, the beginner's solution on the official Rubik's website was an Edges First solution (link). It uses the following five algorithms:
 Edge switcher: (D'FD'F')D2(R'D'RD)BR'B'
 Edge switcher with flip: U2FRF2(UFU')R(BUB')R2U2
 Switch three corners: (R'ULU')(RUL'U')
 Corner flipper right: (LUL'U)(LU2L')(R'U'RU')(R'U2R)
 Corner flipper left: (R'U2R)(UR'UR)(LU2L')(U'LU'L')
The solution steps are:
 Form a toplayer cross
 Position and orient the middle edges (with the "Edge switcher" and the "Edge switcher with flip")
 Position and orient the bottom edges (with the "Edge switcher" and the "Edge switcher with flip")
 Position the corner cubes (with the "Switch three corners" algorithm)
 Orient the corner cubes (with the "Corner flipper" algorithms)